Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1997. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 24)

JONES, J. P. and Kiss, P., Some congruences concerning second order linear recurrences

Some congruences concerning second order linear recurrences 31 which follows from a

3

n - ß

3

n = 3

(a

n - ß

n)a

nß

n + (

a

n - ß

nf. Lemma 2. For any non-negative integers m and n we have U

m+2n = VnUm + n ~ B

nU

m. Proof. Similarly as in the proof of Lemma 1, „,771+271 _ ftm+2n _m+n _ ßm + n ,

vm _ oi = (a

n+/T)~ M)

n y/D y/D ^[D is an identity which by (1) and (2), implies the lemma. Lemma 3. For any n > 0 we have V

2

n = 2B

n + DUl = V

2 - 2B

n and U

2

n = U

nV

n. Proof. The identities /yv

n — fl

n\

2 r\

2

n — fl

2

n rv

n — fl

n a

2

n + ß

2

n = 2(aß)

n + D and " j

* = + ^ prove the lemma. Proof of the Theorem. We prove the first congruence of the Theorem by double induction on k. For k = 1 and k = 3, by Lemma 1, the congruence is an identity. Suppose the congruence holds for k and k -f 2, where k > 1 is odd. Then by Lemma 2 and 3 we have U

n(k+

4) — Unk+in = U

nk+2n ~ B Unk (3) = (2B

n + DU

2

n)U

n{k

+2) ~B

2

nU

n

k = (2

B

n + DUl)Q - B

2

nR (mod D

2U'*), where (4) Q = (k + 2)B^U

n + »)'-!) and (5) R = kB^

nU

n + '

^DB^Ul